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\title[ch05]{Chapter 19: STABILITY OF DIFFERENTIAL EQUATIONS}
\author[]{SCC}
%\institute[XX大学]{XX大学\quad 数学与统计学院\quad 数学与应用数学专业}
%\date{2025年6月}

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% 封面页
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  \titlepage
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% 目录页
\begin{frame}{Contents}
  \tableofcontents
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Section 1
\section{ASYMPTOTIC STABILITY.}
%---------------------------------------------------
\begin{frame}{1.1 DEFINITION. }

%We begin with some basic facts about the stability of singular points of systems of differential equations. 

Consider a differential equation
\begin{equation}
\dot{X} = G(X).
\tag{1.1}
\end{equation}
where
$
G : \mathbb{R}^n \rightarrow \mathbb{R}^n
$
is a function of class $C{\,}^r$ for some $r > 2$, and assume that $G(0) = 0$. 

%By the uniqueness theorem [Arnold 81, Ch. 2, \S8.3], the solution $\phi$ with initial condition $\phi(0) = 0$ is $\phi = 0$. 
%We are interested in the behaviour of solutions with neighbouring initial conditions. 

The singular point $X = 0$ of the equation is said to be {\color{red}asymptotically stable} if:

\begin{enumerate}
\item Given $\epsilon > 0$, there exists $\delta > 0$ (depending only on $\epsilon$ and not on $t$) such that, for every $P_0$ with $|P_0| < \delta$, the solution $\phi$ of (1.1) with initial condition $\phi(0) = P_0$ can be extended to the whole half line $t > 0$ and satisfies $|\phi(t)| < \epsilon$ for every $t > 0$.

\item There exists $\eta > 0$ such that $\lim\limits_{t \to +\infty} \phi(t) = 0$ for all solutions $\phi$ of (1.1) which satisfy $|\phi(0)| < \eta$.
\end{enumerate}

%Condition (1) above means that if the solution is initially within a ball of radius $\delta$ around the origin then it will never leave a ball of radius $\epsilon$. 


\noindent\rule{\textwidth}{0.4pt}

\end{frame}
  
%---------------------------------------------------
\begin{frame}{1.2 THEOREM. }

Asymptotic stability is easy to determine for linear systems.

Let $A$ be an $n \times n$ matrix with entries in $\mathbb{R}$. 

The origin is an {\color{red}asymptotically stable} singular point of $$\dot{X} = A X$$ if and only if all the eigenvalues of $A$ have negative real part.

\noindent\rule{\textwidth}{0.4pt}

\end{frame}
  
%---------------------------------------------------
\begin{frame}{1.3 THEOREM. }

 If the real part of every eigenvalue of $JG(0)$ is negative, then $0$ is an {\color{red}asymptotically stable} point of the differential equation
\begin{equation}
\dot{X} = G(X).
\tag{1.1}
\end{equation}

\noindent\rule{\textwidth}{0.4pt}

\end{frame}
  
%---------------------------------------------------
\begin{frame}{1.4 CONJECTURE. }

We shall say that $0$ is {\color{red}globally asymptotically stable} if $\eta$ may be taken to be $\infty$ in (2) in the definition. 

For a linear system, if $0$ is asymptotically stable, then it is {\color{red}globally asymptotically stable}. 

The origin is {\color{red}globally asymptotically stable} for (1.1) if, for each $P \in \mathbb{R}^n$, the origin is an {\color{red}asymptotically stable} point of the system $\dot{X} = JG(P) \cdot X$.

\noindent\rule{\textwidth}{0.4pt}

\end{frame}
  
%---------------------------------------------------
\begin{frame}{1.5 THEOREM. }

Let $\mathcal{F}$ be the class of $C{\,}^1$ maps $F : \mathbb{R}^2 \rightarrow \mathbb{R}^2$ which satisfy the following properties:

\begin{enumerate}
    \item $F(0) = 0$;
    \item $\text{tr} JF(P) < 0$ for all $P \in \mathbb{R}^2$;
    \item $\det JF(P) > 0$ for all $P \in \mathbb{R}^2$.
\end{enumerate}

Let $F$ be a polynomial function in $\mathcal{F}$ then the origin is a {\color{red}globally asymptotically stable} point of the system $\dot{X} = F(X)$.

\noindent\rule{\textwidth}{0.4pt}

\end{frame}
  
%---------------------------------------------------
\begin{frame}{1.6 PROPOSITION. }

Suppose that the origin is a {\color{red}globally asymptotically stable} point of $\dot{X} = F(X)$ for every polynomial map $F \in \mathcal{F}$. Then the polynomial maps in $\mathcal{F}$ are injective.

\noindent\rule{\textwidth}{0.4pt}

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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Section 2
\section{GLOBAL UPPER BOUND.}
%---------------------------------------------------
\begin{frame}{2.1 CONSTRUCTION. }

Let $F=(F_1,\ldots,F_n): K^n \rightarrow K^n$ be a polynomial map and let $\Delta(x) = \det JF(x)$. 
%Throughout this section we will assume that $\Delta(x) \neq 0$ for every $x \in K^n$. 
%Note that since we are not assuming that $K$ is algebraically closed, this does not imply that $\Delta(x)$ is constant. 
%Put $d = \deg F = \max\{\deg F_i : 1 \leq i \leq n\}$.
Let $g \in K[X,\Delta^{-1}]$ and consider the derivations $D_i$ of $K[X,\Delta^{-1}]$ defined for $i=1,\ldots,n$ by
$$
D_i(g) = \Delta^{-1} \det J(F_1,\ldots,F_{i-1},g,F_{i+1},\ldots,F_n).
$$
%Note that $$D_i = \Delta^{-1} \sum_{k=1}^{n} a_{ik} \partial_k$$ where $a_{ik}$ is the $ik$ cofactor. Hence, $\deg(a_{ik}) \leq (n-1)d$. By Lemma 4.4.1, these derivations satisfy $$ [D_i,F_j] = \delta_{ij} \quad \text{and} \quad [D_i,D_j] = 0 $$ for $1 \leq i,j \leq n$.

We shall use the $D_i$ to define an $A_n$-module structure on $K[X,\Delta^{-1}]$ as follows:
\begin{align*}
x_i \bullet g &= F_i \cdot g, \\
\partial_i \bullet g &= D_i(g),
\end{align*}
where $g \in K[X,\Delta^{-1}]$. 
%A routine argument using Appendix 1 shows that 
Then $\bullet$ gives a well-defined action of $A_n$-module on $K[X,\Delta^{-1}]$. We denote this module by $M(F)$.

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%---------------------------------------------------
\begin{frame}{2.2 LEMMA. }

As an $A_n$-module, $M(F)$ is holonomic and its multiplicity cannot exceed $2^n(2nd+1)^n$, where $d = \deg F = \max\{\deg F_i : 1 \leq i \leq n\}$.


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%---------------------------------------------------
\begin{frame}{2.3 THEOREM. }

Let $F : K^n \rightarrow K^n$ be a polynomial map. If $\det J(F) \neq 0$ everywhere in $K^n$, then there exists a positive integer $b$ such that $F^{-1}(P)$ does not have more than $b^n$ points for every $P \in K^n$.

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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Section 3
\section{GLOBAL STABILITY ON THE PLANE.}
%---------------------------------------------------
\begin{frame}{3.1 THEOREM. }

Let $F \in \mathcal{F}$. If there exist positive constants $\rho$ and $r$ such that
$$
|F(X)| \geq \rho \quad \text{whenever} \quad |X| \geq r
$$

then the origin is a {\color{red}globally asymptotically stable} point of the system $\dot{X} = F(X)$.

\noindent\rule{\textwidth}{0.4pt}

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%---------------------------------------------------
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Section 4
\section{EXERCISES.}
%---------------------------------------------------
\begin{frame}{EXERCISE 1. }

Let $A$ be a $2 \times 2$ matrix with real coefficients. Show that the origin is a {\color{red}globally asymptotically stable} point of the system $\dot{X} = A X$ if and only if the real part of the eigenvalues of $A$ are negative.

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%---------------------------------------------------
\begin{frame}{EXERCISE 2. }

Let $F \in \mathcal{F}$. Show that if $F$ is {\color{red}globally invertible} in $\mathbb{R}^2$ then the origin is a {\color{red}globally asymptotically stable} point of the system $\dot{X} = F(X)$.

\noindent\rule{\textwidth}{0.4pt}

Hint: By the inverse function theorem there exist $\rho, r > 0$ such that $F$ maps $B_\rho(0)$ into $B_r(0)$. Since $F$ is globally one-to-one the points outside $B_\rho(0)$ must be sent outside $B_r(0)$. But this is the hypothesis of Theorem 3.1.

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%---------------------------------------------------
\begin{frame}{EXERCISE 3. }

Let $F \in \mathbb{R}[x,y]$. Use Green's theorem to show that if $\det J(F) = 1$ everywhere on $\mathbb{R}^2$ then $F$ is a map of $\mathbb{R}^2$ that preserves area.

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